I thought they do, but they result in spawning new areas of mathematics. Though more often than disagreements, it is questions are are unanswered and either proven unanswerable, or have failed all attempts to achieve an answer. For example, math has been developed for both the outcome that the Riemann hypothesis is true and that it is false. There is sometimes even the fun possibility of these questions being unsolvable and the implications it has if we assume it will eventually be proven a question cannot be answered yes or no (though I can't recall if there are any significant results from assuming such).

My memory is a bit poor on the matter, and my knowledge limited, but I think I've even read of people disagreeing with fundamental concepts in mathematics and attempting to see what happens when you do so. Does it result in a field of math that behaves the same? Does it make working certain problems easier and others harder?

And then you get the fun stuff like this.

https://en.wikipedia.org/wiki/Intuitionism

https://en.wikipedia.org/wiki/Law_of_excluded_middle#Critici...

But alas, I have too little education to be more than a confused spectator of such debates.